Rewrite the function by completing the square. $f(x)= 4 x^{2} +12 x +9$ $f(x)=$
Answer: $\begin{aligned} f(x)&=4 x^2 +12 x +9 \\\\ &=4 \left(x^2 +3 x\right) +9 \end{aligned}$ Now we want to complete $x^2 +3 x$ into a perfect square. To do that, we should add $\left(\dfrac{{3}}{2}\right)^2={\dfrac{9}{4}}$ to it: $x^2{+3}x+{\dfrac{9}{4}}=\left(x +\dfrac{3}{2}\right)^2$ We add ${\dfrac{9}{4}}$ inside the parentheses, and subtract ${4}\cdot{\dfrac{9}{4}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=}{4} \left(x^2 +3 x\right) +9 \\\\ &={4}\left(x^2 +3 x+{\dfrac{9}{4}}\right) +9 -{4}\cdot{\dfrac{9}{4}} \\\\ &=4 \left(x +\dfrac{3}{2}\right)^2 +9 -9 \\\\ &=4 \left(x +\dfrac{3}{2}\right)^2 +0 \end{aligned}$ In conclusion, the function after completing the square is written as: $f(x)=4 \left(x +\dfrac{3}{2}\right)^2 +0$